Generalized Frobenius partitions, Motzkin paths, and Jacobi forms

TL;DR

This paper connects generalized Frobenius partitions to Jacobi forms, providing explicit formulas, new congruences, and combinatorial interpretations through Motzkin paths, enriching the understanding of their modular properties.

Contribution

It reformulates generating functions for generalized Frobenius partitions as Jacobi form coefficients, leading to explicit formulas and new combinatorial and congruence results.

Findings

Explicit formulas for partition generating functions in terms of q-products

New congruences for generalized Frobenius partitions

Combinatorial interpretation via Motzkin paths

Abstract

We show how Andrews' generating functions for generalized Frobenius partitions can be understood within the theory of Eichler and Zagier as specific coefficients of certain Jacobi forms. This reformulation leads to a recursive process which yields explicit formulas for the generalized Frobenius partition generating functions in terms of infinite $q$-products. In particular, we show that specific examples of our result easily reestablish previously known formulas, and we describe new congruences, both conjectural and proven, in additional cases. The modular structure of Jacobi forms indicates that \emph{all} of the coefficients of the forms are of interest. We give a combinatorial definition of these "companion series" and explore their combinatorics via the counting of Motzkin paths.